We consider two symmetry metrics to detect partisan gerrymandering: the Mean-Median Difference (MM) and Partisan Bias (PB). To lay the groundwork for our main results, we first assert that the foundation of a partisan gerrymander is to draw a map so that the preferred party wins an extreme number of seats, and that both the Mean-Median Difference and Partisan Bias have been used to detect partisan gerrymandering. We then provide both a theoretical and empirical analysis of the Mean-Median Difference and Partisan Bias. In our theoretical analysis, we consider vote-share, seat-share pairs (V,S) for which one can construct election data having vote share V and seat share S, and turnout is equal in each district. We calculate the range of values that MM and PB can achieve on that constructed election data. In the process, we find the range of vote-share, seat share pairs (V,S) for which there is constructed election data with vote share V , seat share S, and MM = 0, and see that the corresponding range for PB is the same set of (V,S) pairs. We show how the set of such (V,S) pairs allowing for MM = 0 (and PB = 0) changes when turnout in each district is allowed to be different. By observing the results of this theoretical analysis, we give examples of how these two metrics are unable to detect when a map has an extreme number of districts won. Because these examples are constructed, we follow this with our empirical study, in which we show on 18 different U.S. maps that these two metrics are unable to detect when a map has an extreme number of districts won.

翻译：我们考虑两种用于检测党派不公正划分选区的对称性指标：均值-中位数差值与党派偏差。为奠定主要结果的基础，我们首先断言，党派不公正划分选区的根本目的是绘制使偏好政党赢得极端数量席位的选区地图，而均值-中位数差值与党派偏差均已被用于检测此类行为。随后，我们对均值-中位数差值与党派偏差进行了理论与实证分析。在理论分析中，我们考虑得票率-席位率配对(V,S)，针对此类配对可以构建具有得票率V和席位率S的选举数据，且各选区投票率相等。我们计算了在此构建的选举数据上MM和PB所能达到的数值范围。在此过程中，我们找出了存在构建选举数据使得得票率为V、席位率为S且MM=0的(V,S)配对范围，并发现PB对应的范围与此相同。我们展示了当允许各选区投票率不同时，允许MM=0（及PB=0）的此类(V,S)配对集合如何变化。通过观察理论分析结果，我们举例说明了这两种指标如何无法检测出地图赢得极端数量选区的情况。由于这些案例是构建的，我们随后进行了实证研究，通过对18张不同的美国选区地图的分析，证明这两种指标确实无法检测出地图赢得极端数量选区的情况。